Vibrational from diatomic molecules

Figure C3.5.7. Possible modes of vibrational wavepacket (smootli Gaussian curve) motion for a highly vibrationally excited diatomic molecule produced by photodissociation of a linear triatomic such as Hglj, from [8].

The Vibration of Diatomic Molecules.—In addition to their rotation, we have seen that diatomic molecules can vibrate with simple harmonic motion if the amplitude is small enough. We shall use only this approximation of small amplitude, and our first stop will be to calculate the frequency of vibration. To do this, we must first find the linear restoring force when the interatomic distance is displaced slightly from its equilibrium value / ,. We can get this from Eq. (1.2) by expanding the force in Taylor s series in (r — rt). We have... 

The author became interested in collision-induced state-to-state processes as a result of experiments in which narrow-line, circularly polarised laser radiation was used to excite fluorescence from diatomic molecules in collision cells. The laser excites known multipolar moments of nij distribution among specific j states of an excited vibronic state. Wavelength and polarisation analysis of the emission allowed the population and orientation multipoles of nij among states accessed by elastic and inelastic collisions to be determined from measured intensity and degree of circular polarisation. An example is shown in Fig. 1, from which it is evident that the initial asymmetric nij distribution is maintained through elastic and inelastic collisions [13] even when substantial amounts of vibrational and rotational energy... 

The vibration of diatomic molecules may be related to the motion of two masses joined by a spring of negligible mass Tq in length and a force constant of k, such as a simple harmonic oscillator. A small displacement of the particle from its equilibrium position will require a restoring force that obeys Hooke s law ... 

The Schrodinger equation for this two-body problem starts out the same as the general two-body Schrodinger equation (Equation 9.18) however, the potential function, V(r), is different from that of the vibrating-rotating diatomic molecule. It is an electrostatic attraction of two point charges, and its form is... 

Although a diatomic molecule can produce only one vibration, this number increases with the number of atoms making up the molecule. For a molecule of N atoms, 3N-6 vibrations are possible. That corresponds to 3N degrees of freedom from which are subtracted 3 translational movements and 3 rotational movements for the overall molecule for which the energy is not quantified and corresponds to thermal energy. In reality, this number is most often reduced because of symmetry. Additionally, for a vibration to be active in the infrared, it must be accompanied by a variation in the molecule s dipole moment. 

Diatomic molecules have only one vibrational mode, but VER mechanisms are paradoxically quite complex (see examples C3.5.6.1 and C3.5.6.2). Consequently there is an enonnous variability in VER lifetimes, which may range from 56 s (liquid N2 [18]) to 1 ps (e.g. XeF in Ar [25]), and a high level of sensitivity to environment. A remarkable feature of simpler systems is spontaneous concentration and localization of vibrational energy due to anhannonicity. Collisional up-pumping processes such as... 

Since depends on nuclear coordinates, because of the term, so do and but, in the Bom-Oppenheimer approximation proposed in 1927, it is assumed that vibrating nuclei move so slowly compared with electrons that J/ and involve the nuclear coordinates as parameters only. The result for a diatomic molecule is that a curve (such as that in Figure 1.13, p. 24) of potential energy against intemuclear distance r (or the displacement from equilibrium) can be drawn for a particular electronic state in which and are constant. 

As is the case for diatomic molecules, rotational fine structure of electronic spectra of polyatomic molecules is very similar, in principle, to that of their infrared vibrational spectra. For linear, symmetric rotor, spherical rotor and asymmetric rotor molecules the selection mles are the same as those discussed in Sections 6.2.4.1 to 6.2.4.4. The major difference, in practice, is that, as for diatomics, there is likely to be a much larger change of geometry, and therefore of rotational constants, from one electronic state to another than from one vibrational state to another. 

We have seen in Section 6.1.3.2 that, for diatomic molecules, vibrational energy levels, other than those with v = 1, in the ground electronic state are very often obtained not from... 

Vibrational Energy Levels A diatomic molecule has a single set of vibrational energy levels resulting from the vibration of the two atoms around the center of mass of the molecule. A vibrating molecule is usually approximated by a harmonic oscillatord for which... 

Vibrational Contribution to the Gibbs Free Energy For a Linear Diatomic Molecule From equations (10.84) and (10.101)... 

Molecules possess discrete levels of rotational and vibrational energy. Transitions between vibrational levels occur by absorption of photons with frequencies v in the infrared range (wavelength 1-1000 p,m, wavenumbers 10,000-10 cm , energy differences 1240-1.24 meV). The C-0 stretch vibration, for example, is at 2143 cm . For small deviations of the atoms in a vibrating diatomic molecule from their equilibrium positions, the potential energy V(r) can be approximated by that of the harmonic oscillator ... 

The molecular constants o , B, Xe, D, and ae for any diatomic molecule may be determined with great accuracy from an analysis of the molecule s vibrational and rotational spectra." Thus, it is not necessary in practice to solve the electronic Schrodinger equation (10.28b) to obtain the ground-state energy o(R). 

Perhaps the first evidence for the breakdown of the Born-Oppenheimer approximation for adsorbates at metal surfaces arose from the study of infrared reflection-absorption line-widths of adsorbates on metals, a topic that has been reviewed by Hoffmann.17 In the simplest case, one considers the mechanism of vibrational relaxation operative for a diatomic molecule that has absorbed an infrared photon exciting it to its first vibrationally-excited state. Although the interpretation of spectral line-broadening experiments is always fraught with problems associated with distinguishing... 

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